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I've got a pretty odd error on a project I'm working on and was hoping to enlist some advice to fix it. The goal of this notebook is to show that I can eliminate the non-normalizable (blowing up part) of the Whittaker W function for small arguments by multiplying by a constant.

I first start by defining the W function in a more convenient way:

f[r_, k_, w_, v_] := WhittakerW[-I*(k^2/(4*w)), v, -I*r^2*w]

Expanding f (Series[f[r, k, w, v], {r, 0, 1}, Assumptions -> {r>0, w>0, v>0, k>0}]) I see that the constant I need to multiply by is $c = \frac{\Gamma[1/2 + \nu + i(k^2/(4*\omega))]}{(-i)^{1/2 - \nu}}$. This is the imaginary term on all of the non-normalizable parts of $W$ (you can see this if you include higher order terms in the series expansion). So multiplying by this should, in principle, wipe out the imaginary part of the non-normalizable piece, and leave me with a normalizable imaginary part. Unfortunately, it isn't working out this way in Mathematica:

const[k_,w_,v_] = Gamma[1/2 + v + I*(k^2/(4*w))]/(-I)^(1/2 - v)
LogLogPlot[Abs[Im[const[10, .01, 3.1]*f[r, 10, .01, 3.1]]], {r, 0.001, .01}]

The above plot shows a line with negative slope, going to infinity, rather than zero. When I define a function for the Taylor series, however:

bdry2[r_] = 
 Normal[FullSimplify[
   Series[const[10, .01, 3.1]*f[r, 10, .01, 3.1], {r, 0, 5}, 
    Assumptions -> r > 0]]]

The plot: LogLogPlot[{Abs[Im[bdry2[r]]]}, {r, 0.000001, .001}] is normalizable. So this is quite confusing for me, and I think there is some sort of rounding error going on -- but I'm not quite sure where my error lies. Any help would be much appreciated.

Thanks in advance!

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closed as unclear what you're asking by user9660, MarcoB, RunnyKine, Öskå, Jens May 6 '16 at 15:51

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    $\begingroup$ I believe it is a precision issue, try this to force high precision calculation: N[const[10, 1/100, 31/10]*f[2/1000, 10, 1/100, 31/10], 100] you will see the result is purely real . $\endgroup$ – george2079 Aug 25 '15 at 15:52
  • $\begingroup$ Ok - interesting. It seems the imaginary part is 10^(-84) here -- whereas without including as much precision, it was probably non-negligible. Any idea how I can implement this level of precision into my plotting? $\endgroup$ – Schwinger Aug 25 '15 at 15:55
  • $\begingroup$ I suspect this requires more of a mathematics study of the function rather than relying on numerical evaluation even at high precision. $\endgroup$ – george2079 Aug 25 '15 at 16:01
  • $\begingroup$ I think I have good reason to believe what I'm doing is possible numerically -- there are only a couple terms in the taylor series that are non-normalizable (for small v, but still this should hold for any v) -- and they all have some factor $c*r^{-\gamma}$, where $c$ is complex -- so dividing by c should make the coefficient of that term real. Not quite sure I understand the difficulty Mathematica's having with it $\endgroup$ – Schwinger Aug 25 '15 at 16:08
  • $\begingroup$ Schwinger, you should be able to implement increased precision in your plot using arbitrary precision arguments as mentioned by @george2079, by specifying the plotting range in the same way (i.e. {r, 1/1000, 1/100}), and finally by using the WorkingPrecision option to the plotting function. For instance, you could use WorkingPrecision -> 100. $\endgroup$ – MarcoB Mar 9 '16 at 4:11